Finite Group Databasebeta

← #505664

Subgroups — #505664

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2236C212yesmaximal
2C4×#467C211yesmaximal, characteristic
3C2×#1845C211yesmaximal, characteristic
4#13505C212yesmaximal, complemented
5#13481C212yesmaximal, complemented
6#13176C212yesmaximal, complemented
7#12867C212yesmaximal
8#12812C212yesmaximal, complemented
9#12783C211yesmaximal, characteristic
10C2×#467C2211yescharacteristic
11C2×C4×Q16C2211yescharacteristic
12C22×#107—21no—
13C22×#107C2212yes—
14C22×C4×C8C2211yescharacteristic
15C4×C22⋊Q8—21no—
16C2×#384—21no—
17#2236—28no—
18C2×#366C2214yes—
19C2×C4⋊C16C2212yes—
20C2×#358C2214yes—
21C2×C22⋊C16C2212yes—
22#1969C2211yescharacteristic
23#1942C2211yescharacteristic
24#1933—21no—
25#1892—21no—
26#1845—21no—
27#1845—21no—
28#1845C2212yes—
29#1845C2212yes—
30#1808C2211yescharacteristic
31C4×#107C2211yescharacteristic
32C4×#103—21no—
33#1380C2214yes—
34#1380C2214yes—
35#1374C2212yes—
36C22×Q16C2311yescharacteristic
37C23×C8D411yescharacteristic
38C23×C8C2311yescharacteristic
39C2×C22⋊Q8—21no—
40C2×C4×Q8—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—41no—
43C22×C4⋊C4—41no—
44C22×C42D411yescharacteristic
45C22×C16C2314yes—
46#467—22no—
47#467D414yes—
48C4×Q16—24no—
49C2×#107—24no—
50C2×#107—24no—
51C2×#107—22no—
52C2×#107—22no—
53C2×#107—22no—
54C2×#107—44no—
55C2×#107C2311yescharacteristic
56C2×#107C2311yescharacteristic
57C2×#107C2314yes—
58C2×#107C2311yescharacteristic
59C2×#103—21no—
60C2×#103—21no—
61C2×#103—21no—
62C2×C4×C8—24no—
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#391—21no—
68#389—21no—
69#387—41no—
70#385—41no—
71#384—22no—
72#384—22no—
73#384—41no—
74#384—41no—
75#380—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—41no—
78C4×C22⋊C4—41no—
79#366—28no—
80C4⋊C16—24no—
81#358—28no—
82C22⋊C16—24no—
83#340—21no—
84#340—21no—
85#340—21no—
86C22×Q8—21no—
87C23×C4—42no—
88C23×C4D811yescharacteristic
89C23×C4D811yescharacteristic
90C23×C4C2×D411yescharacteristic
91C2×Q16—22no—
92C2×Q16—21no—
93C2×Q16—22no—
94C2×Q16C2×D412yes—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8#4112yes—
105C22×C8C2411yescharacteristic
106C22⋊Q8—24no—
107C22⋊Q8—42no—
108C4×Q8—42no—
109C4×Q8—42no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—84no—
125C2×C4⋊C4—84no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42D811yescharacteristic
133C2×C42D811yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C16—24no—
137C2×C16—24no—
138C2×C16—24no—
139#107—21no—
140#107—24no—
141#107—24no—
142#107—22no—
143#107—22no—
144#107—44no—
145#107—44no—
146#107—44no—
147#107C2×D412yes—
148#103—24no—
149#103—42no—
150#103—42no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8#4114yes—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160C24C2×D811yescharacteristic
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—44no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4Q3214yes—
185C22×C4#12112yes—
186C22×C4C2×D811yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4C2×D811yescharacteristic
189C22×C4C2×D811yescharacteristic
190C22×C4C2×D811yescharacteristic
191C22×C4AGL(1,ℤ/8)12yes—
192C22×C4C22×D411yescharacteristic
193Q16—24no—
194Q16—42no—
195C2×C8—28no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8#12112yes—
205C2×C8#12312yes—
206C2×C8#13712yes—
207C2×C8C22×D411yescharacteristic
208C2×C8C2×#4112yes—
209C2×C8C2×#4111yescharacteristic
210C2×C82−1+412yes—
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—44no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—84no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C22⋊C4—44no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—22no—
232C42—22no—
233C42—82no—
234C42—82no—
235C42#12312yes—
236C42#13712yes—
237C16—44no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23#45912yes—
243C23C2×Q3212yes—
244C23#51012yes—
245C23C22×D811yescharacteristic
246Q8—42no—
247Q8—42no—
248Q8—81no—
249Q8—81no—
250C2×C4—28no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—84no—
279C2×C4—168no—
280C2×C4#45912yes—
281C2×C4#48012yes—
282C2×C4#48012yes—
283C2×C4C2×Q3212yes—
284C2×C4C2×Q3212yes—
285C2×C4#50912yes—
286C2×C4#50912yes—
287C2×C4C2×#12111yescharacteristic
288C2×C4C2×#12311yescharacteristic
289C2×C4#53611yescharacteristic
290C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
291C2×C4C2×#13711yescharacteristic
292C2×C4C2×#13711yescharacteristic
293C8—22no—
294C8—22no—
295C8—22no—
296C8#54712yes—
297C8#55212yes—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#220914yes—
305C22C2×#45911yescharacteristic
306C22C2×#48011yescharacteristic
307C22#322611yescharacteristic
308C22C22×Q3211yescharacteristic
309C22C2×#50911yescharacteristic
310C22C2×#50911yescharacteristic
311C22C2×#51011yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—82no—
317C4—82no—
318C4—161no—
319C4—164no—
320C4—161no—
321C4#223614yescomplemented
322C4#248611yescharacteristic
323C4#303911yescharacteristic
324C4#309211yescharacteristic
325C4#323811yescharacteristic
326C2—24no—
327C2#2764411yescharacteristic
328C2C2×#220911yescharacteristic
329C2#3358211yescharacteristic
330C2C2×#223611yescharacteristic
331C2#3364611yescharacteristic
332C2#3379611yescharacteristic
333C2#3380511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #13505C2semidirect
5. #13481C2semidirect
6. #13176C2semidirect
8. #12812C2semidirect
321. C4#2236semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#22362
C4×#4672
C2×#18452
#135052
#134812
#131762
#128672
#128122
#127832